Standard +0.3 Part (a) is straightforward translation into a differential equation requiring only understanding of proportionality. Part (b)(i) involves separable variables with substitution (u = 2x-1) and integration requiring some algebraic manipulation, but follows standard C4 techniques. Part (b)(ii) is simple substitution. Slightly above average due to the algebraic complexity in integration, but still a routine C4 question.
A water tank has a height of 2 metres. The depth of the water in the tank is \(h\) metres at time \(t\) minutes after water begins to enter the tank. The rate at which the depth of the water in the tank increases is proportional to the difference between the height of the tank and the depth of the water.
Write down a differential equation in the variables \(h\) and \(t\) and a positive constant \(k\).
(You are not required to solve your differential equation.)
Another water tank is filling in such a way that \(t\) minutes after the water is turned on, the depth of the water, \(x\) metres, increases according to the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 15 x \sqrt { 2 x - 1 } }$$
The depth of the water is 1 metre when the water is first turned on.
Solve this differential equation to find \(t\) as a function of \(x\).
Calculate the time taken for the depth of the water in the tank to reach 2 metres, giving your answer to the nearest 0.1 of a minute.
(l mark)
8
\begin{enumerate}[label=(\alph*)]
\item A water tank has a height of 2 metres. The depth of the water in the tank is $h$ metres at time $t$ minutes after water begins to enter the tank. The rate at which the depth of the water in the tank increases is proportional to the difference between the height of the tank and the depth of the water.
Write down a differential equation in the variables $h$ and $t$ and a positive constant $k$.\\
(You are not required to solve your differential equation.)
\item \begin{enumerate}[label=(\roman*)]
\item Another water tank is filling in such a way that $t$ minutes after the water is turned on, the depth of the water, $x$ metres, increases according to the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 15 x \sqrt { 2 x - 1 } }$$
The depth of the water is 1 metre when the water is first turned on.\\
Solve this differential equation to find $t$ as a function of $x$.
\item Calculate the time taken for the depth of the water in the tank to reach 2 metres, giving your answer to the nearest 0.1 of a minute.\\
(l mark)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2012 Q8 [12]}}